1. Field of the Disclosure
This disclosure relates generally to multiphase flow measurements of wellbore fluids.
2. Description of the Related Art
Wellbore fluids often are multi-phase fluids that contain oil, gas and water. The composition, flow rate, and viscosity of each component (oil, water, and gas) vary from well to well. Usually, the flow rate of the gas is the fastest and that of the oil the slowest, unless the fluid is well-mixed and gas is entrained inside the liquid. The large variety of flow patterns in which the liquid and gas might be distributed and the variations in the physical properties of each component makes flow rate prediction of each component difficult.
At low fluid velocities, wellbore liquids tend to accumulate at low pockets in horizontal pipes while gas coalesces into large and small bubbles, which propagate faster than the liquid in vertical tubings or risers. Both of these aspects cause the gas to flow faster than the liquid, or in other words, increase the slip between the gas and liquid. Fluid density is a parameter used for determining the flow of multiphase fluids. Some methods utilize spot density, which is the density at a particular cross-section of the flow conduit, over a very narrow (compared to the hydraulic diameter) length of the conduit. Spot density may be different from the homogeneous mixture density due to the slip between the gas and liquid in the multiphase fluid. A common method to measure the spot density of a fluid utilizes a radioactive source and measures the absorption of the gamma rays by the fluid media. This method is more sensitive to minerals than to the hydrocarbons and the density is typically measured across a small section of the pipe carrying the fluid. Because of this, spot density measurement, information regarding slip and thus the corrected bulk-density is lost. Bulk density, which is the density of the fluid mixture in a reasonably long flow conduit (compared to the hydraulic diameter), is more representative of the average density. Bulk density requires less slip correction but is still dependent on slip.
Mathematical models have been used for computing multi-phase fluid flow. Such methods, however, require rigorous knowledge of the boundary conditions of multiple parameters, such as surface tension, viscosity, fluid mixture, etc. As such parameters are not be measured in line (in-situ), the value of slip is assumed or obtained from certain empirical experiments. This confines the validity of the mathematical model to the specific assumptions made or the results of the experiments made. The slip value in the multiphase fluid produced from a wellbore is sometimes different from such experimentally determined slip values, and thus large errors can result. We illustrate the above statement by way of an example. FIGS. 1a and 1b illustrate two typical flow regimes in horizontal and vertical pipe flows respectively. If the inclination of the horizontal pipe is changed slightly to +15° upward or −15° downward, the flow pattern will be completely different from what is shown. Similarly, the flow pattern for an inclination of 5° will be different from that for an inclination of 15°, 11°, etc. and therefore the resulting slip values will be completely different. Since there is a huge variety of piping configurations and fluid parameter values, using empirically determined slip values will lead to large errors.
The flow of a multiphase wellbore fluid may be expressed as a set of non-linear partial differential equations, as given below:
Conservation of Mass:
                                                        ∂                              ∂                t                                      ⁢                          (                                                ρ                  N                                ⁢                                  α                  N                                            )                                +                                    ∂                              (                                                      ρ                    N                                    ⁢                                      j                    Ni                                                  )                                                    ∂                              x                i                                                    =                  I          N                                    Eqn        .                                  ⁢        1            Conservation of Momentum:
                                                        ∂                              ∂                t                                      ⁢                          (                                                ρ                  N                                ⁢                                  α                  N                                ⁢                                  u                  Nk                                            )                                +                                    ∂                              ∂                                  x                  i                                                      ⁢                          (                                                ρ                  N                                ⁢                                  α                  N                                ⁢                                  u                  Ni                                ⁢                                  u                  Nk                                            )                                      =                                            α              N                        ⁢                          ρ              N                        ⁢                          g              k                                +                      F            Nk                    -                                    δ              N                        ⁢                          {                                                                    ∂                    p                                                        ∂                                          x                      k                                                                      -                                                      ∂                                          σ                      Cki                      D                                                                            ∂                                          x                      i                                                                                  }                                                          Eqn        .                                  ⁢        2            Conservation of Energy
                                                        ∂                              ∂                t                                      ⁢                          (                                                ρ                  N                                ⁢                                  α                  N                                ⁢                                  e                  N                  *                                            )                                +                                    ∂                              ∂                                  x                  i                                                      ⁢                          (                                                ρ                  N                                ⁢                                  α                  N                                ⁢                                  e                  N                  *                                ⁢                                  u                  Ni                                            )                                      =                              Q            N                    +                      W            N                    +                      E            N                    +                                    δ              N                        ⁢                          ∂                              ∂                                  x                  k                                                      ⁢                          (                                                u                  Ci                                ⁢                                  σ                  Cij                                            )                                                          Eqn        .                                  ⁢        3            
In the above equations, the subscript N denotes a specific phase or component, which in the case of wellbore fluid may be oil (O), water (W) and gas (G). The lower case subscripts (i, ik, etc.) refer to vector or tensor components. We follow the tensor notation where a repeated lower case subscript implies summation over all of its possible values, e.g.uiui=u1u1+u2u2+u3u3  Eqn. 4ρN is the density of component N, αN is the volume fraction of component N, and jNi is the volumetric flux (volume flow per unit area) of component N, where i is 1, 2, or 3 respectively for one-dimensional, two-dimensional or three-dimensional flow. IN results from the interaction of different components in the multiphase flow. IN is the rate of transfer of mass to the phase N, from the other phases per unit volume. uNk is the velocity of component N along direction k. The volumetric flux of a component N and its velocity are related by:jNk=αNuNk  Eqn. 5gk is the direction of gravity along direction k, p is the pressure, σCkiD is the deviatoric component of the stress tensor σCki acting on the continuous phase, FNk is the force per unit volume imposed on component N by other components within the control volume.
e*N is the total internal energy per unit mass of the component N.
Therefore,
                              e          N          *                =                              e            N                    +                                    1              2                        ⁢                          u              Ni                        ⁢                          u              Ni                                +          gz                                    Eqn        .                                  ⁢        6            where eN is the internal energy of component N. QN is the rate of heat addition to component N from outside the control volume, WN is the rate of work done to N by the exterior surroundings, and EN is the energy interaction term, i.e. the sum of the rates of heat transfer and work done to N by other components within the control volume.
The above equations are subject to the following constraints:
                                          ∑            N                    ⁢                      I            N                          =        0                            Eqn        .                                  ⁢        7                                                      ∑            N                    ⁢                      F            Nk                          =        0                            Eqn        .                                  ⁢        8                                                      ∑            N                    ⁢                      E            N                          =        0                            Eqn        .                                  ⁢        9            
The above equations are a system of nonlinear partial differential equations, the solution to which, when it exists for a set of narrow initial and boundary conditions, results in a series which does not always converge. Also, since the initial conditions and boundary conditions, such as the initial bubble size and the distribution of bubbles in the conduit are neither measured nor known a priori for the different flow regimes, those conditions are thus often estimated from prior historical knowledge. Since the above methods make certain assumptions, they may lead to inaccurate results.
The disclosure herein provides an improved apparatus and method for multiphase fluid measurements.